Application of the Multistage Telescoping Decomposition Method for Solving Fractional Logistic Equations

The classical logistic equation does not account for memory effects, motivating fractional formulations for modelling systems with memory and hereditary properties. This study applies the Multistage Telescoping Decomposition Method (MTDM) to solve fractional logistic differential equations (FLDEs) formulated using the Caputo fractional derivative. The FLDE was reformulated, and MTDM was employed to construct semi-analytical approximate solutions without requiring Adomian polynomials or Lagrange multipliers. Numerical examples were used to evaluate accuracy, convergence, and computational efficiency across different fractional orders. The resulting approximations were compared with those obtained using the Telescoping Decomposition Method (TDM), New Iterative Method (NIM), and Residual Power Series Method (RPSM). Error analysis and graphical comparisons indicated that MTDM produced highly accurate approximations with faster convergence and reduced computational complexity in the examined examples. These findings support MTDM as an efficient semi-analytical alternative for solving FLDEs and indicate its potential applicability to broader classes of nonlinear fractional differential equations. The study contributes a computational approach to fractional logistic modelling, with potential applications in applied mathematics, physics, and biological modelling.